One side of a rectangle lies along the line 4x + 7y + 5 = 0. Two of its vertices are (- 3, 1) and (1, 1). Which of the following may be an equation which represents any of the other three straight lines?

Options

A. 7 x - 4y = 3
B. 7x - 4y + 3 = 0
C. y + 1 = 0
D. 4x + 7y = 3

Answer

To solve the problem, we need to find the equations of the remaining sides of the rectangle given that one side lies along the line \(4x + 7y + 5 = 0\) and two vertices of the rectangle are at the points \((-3, 1)\) and \((1, 1)\).

### Step-by-step Solution:

1. **Identify the Given Line**:
The equation of one side of the rectangle is given as:
\[ 4x + 7y + 5 = 0 \]

2. **Identify the Vertices**:
The two vertices of the rectangle are:
\[ A(-3, 1) \quad \text{and} \quad B(1, 1) \]

3. **Determine the Orientation of the Rectangle**:
Since \(A\) and \(B\) have the same \(y\)-coordinate, the line segment \(AB\) is horizontal. Therefore, the sides \(AD\) and \(BC\) must be vertical.

4. **Find the Equation of Line \(AB\)**:
The slope of line \(AB\) is \(0\) (horizontal line), and since it passes through point \(A(-3, 1)\), we can express the equation of line \(AB\) as:
\[ y = 1 \]
This can also be written in standard form as:
\[ 0x + 1y - 1 = 0 \quad \text{or simply} \quad y - 1 = 0 \]

5. **Find the Equation of Line \(AD\)**:
The line \(AD\) is perpendicular to \(AB\) and lies along the line given in the problem. Since \(AD\) is along the line \(4x + 7y + 5 = 0\), we can use this equation directly.

6. **Find the Equation of Line \(BC\)**:
Since \(BC\) is parallel to \(AD\), it will have the same coefficients for \(x\) and \(y\) as the line \(AD\). Thus, the equation of line \(BC\) can be expressed as:
\[ 4x + 7y + \lambda = 0 \]
To find \(\lambda\), we substitute the coordinates of point \(B(1, 1)\):
\[ 4(1) + 7(1) + \lambda = 0 \implies 4 + 7 + \lambda = 0 \implies \lambda = -11 \]
Therefore, the equation of line \(BC\) is:
\[ 4x + 7y - 11 = 0 \]

7. **Find the Equation of Line \(DC\)**:
Line \(DC\) is also perpendicular to line \(AB\) and can be expressed in a similar manner. Since it is also perpendicular to \(AD\), we can use the same approach:
\[ 7x - 4y + \mu = 0 \]
To find \(\mu\), we substitute the coordinates of point \(C(1, 1)\):
\[ 7(1) - 4(1) + \mu = 0 \implies 7 - 4 + \mu = 0 \implies \mu = -3 \]
Therefore, the equation of line \(DC\) is:
\[ 7x - 4y - 3 = 0 \]

### Final Equations:
- The equations of the sides of the rectangle are:
- \(AB: y - 1 = 0\)
- \(AD: 4x + 7y + 5 = 0\)
- \(BC: 4x + 7y - 11 = 0\)
- \(DC: 7x - 4y - 3 = 0\)

### Conclusion:
Among the options provided, the equation that represents one of the sides of the rectangle is:
\[ 7x - 4y - 3 = 0 \]